Univ Toronto, Nov 4, 2009

Topological Insulators

J. G. Checkelsky, Y.S. Hor, D. Qu, Q. Zhang, R. J. Cava, N.P.O.Princeton University

1. Introduction 2. Angle resolved photoemission (Hasan)3. Tentative transport signatures 4. Giant fingerprint signal5. Insulator and Superconductor

cond. band

valence band

cond. band

Top Bottom

Conventional insulator Topological insulator

ks

s

Surface states are helical (spin locked to k)Large spin-orbit interactn

Surface state hasDirac dispersion

kxky

Fu, Kane ’06Zhang et al. ’06Moore Balents ‘06Xi, Hughes, Zhang ‘09

crystal

s

Surface statesmay cross gap

A new class of insulators

cond. band

valence band

cond. band

valence band

1. Time-reversal invariance prevents gap formation at crossing

cond. band

?

Violates TRI

cond. band

2. Suppression of 2kF scattering

Spinor product kills matrix elementLarge surface conductance?

2D Fermisurface

Under time reversal(k↑) (-k↓)

sk

Protection of helical states

Kane, Mele, PRL ‘05valence band valence band

Helicity and large spin-orbit coupling

• Spin-orbit interaction and surface E field effectv B = v E in rest frame

• spin locked to B

• Rashba-like Hamiltonian

Like LH and RH neutrinos indifferent universes

vE

B

vE

B

spin aligned with B inrest frame of moving electron

s

s

k

k

Helical, massless Dirac states with opposite chirality on opp.surfaces of crystal

skn̂ FvH

A twist of the mass (gap)

Doped polyacetylene (Su, Schrieffer, Heeger ‘79)

xe /2Domain wall (soliton)traps ½ charge

Mobius strip

1. Gap-twist produces domain wall2. Domain wall traps fractional charge3. Topological (immune to disorder)

Mobius strip like

H

H H

pvx

xpvH

)(

)(*

m(x)

Jackiw Rebbi, PRD ‘76Goldstone Wilczek, PRL ‘81Callan Harvey, Nuc Phys B ‘85Fradkin, Nuc Phys B ‘87D. Kaplan, Phys Lett B ’92

)(

)(

xmp

pxmH

Dirac modes on domain walls of mass field

Chiral zero-energy modeDomain-wall fermion

vacuumTopologicalinsulator

Chiral surface states?

Dirac fermions as domain wall excitation

z

k

QFT with background mass-twist field

x

Callan-Harvey: Domain walls exchange chiral current to solve anomaly problm

Bi SbBi1-xSbx

Fu Kane prediction of topological insulator

z

kMass twist

Mass twist traps surfaceWeyl fermions

Fu, Kane, PRB ‘06

ARPES confirmationHsieh, Hasan, Cava et al.Nature ‘08

Confirm 5 surf states in BiSb

20 eV photons

+-

Angle-resolved photoemission spectroscopy (ARPES)

velocity selector

E

k||

Inte

nsity

E

quasiparticlepeak

Hsieh, Hasan, Cava et al. Nature 2008

ARPES of surface states in Bi1-x Sbx

ARPES results on Bi2Se3 (Hasan group)

Se defect chemistry difficult to control for small DOS

Xia, Hasan et al. Nature Phys ‘09

Large gap ~ 300meV

As grown, Fermi level in conduction band

Photoemission evidence for Topological Insulators

Why spin polarized?

Rashba term on surface

What prevents a gap?

Time Reversal Symmetry

What is expected from transport?

•No 2 kF scattering

•SdH

•Surface QHE (like graphene except ¼)

•Weak anti-localization

Hsieh, Hasan et al., Nature ‘09

Bi2Se3: Typical Transport

Roughly spherical Fermi surface (period changes by ~ 30%)

Metallic electron pocket with mobility ~ 500-5000 cm2/Vs

Carrier density ~ 1019 e-/cm3

Quantum oscillations of Nernst in metallic Bi2Se3

Major problem confrontg transport investigationAs-grown xtals are always excellent conductors,lies in conduction band (Se vacancies).

(1 K) ~ 0.1-0.5 mcm, n ~ 1 x 1018 cm-3

m* ~ 0.2, kF ~ 0.1 Å-1

Fall into the gap

Decreaseelectron density

Solution:Tune by Ca doping

cond. band

valence band

electrondoped

holedoped

targetHor et al., PRB ‘09Checkelsky et al., arXiv/09

Resistivity vs. Temperature : In and out of the gap

Onset of non-metallic behavior ~ 130 K

SdH oscillations seen in both n-type and p-type samples

Non-metallic samples show no discernable SdH

Checkelsky et al., arXiv:0909.1840

Magnetoresistance of gapped Bi2Se3

Logarithmicanomaly

Conductancefluctuations

Giant, quasi-periodic, retraceable conductance fluctuations

Checkelsky et al., arXiv:0909.1840

Magneto-fingerprints

Giant amplitude(200-500 X too large)

Retraceable(fingerprints)

Spin degreesInvolved in fluctuations

Fluctuations retraceableCheckelsky et al., arXiv:0909.1840

Angular Dependence of R(H) profile Cont.

For δG, 29% spin term

For ln H, 39% spin term (~200 e2/h total)

Theory predicts both to be ~ 1/2π

(Lee & Ramakrishnan), (Hikami, Larkin, Nagaoka)

Conductance -- sum over Feynman paths

ji

i

jii

iji

jijieAAAAAG

,

)(

,

* ||||2

Universal conductance fluctuations (UCF)

G = e2/h

Universal Conductance Fluctuations

in a coherent volume defined by thermal length LT = hD/kT

At 1 K, LT ~ 1 m

For large samples size L, 212 /

L

L

h

eG T

H

LT

Stone, Lee, Fukuyama (PRB 1987)

LT

L = 2 mm“Central-limit theorem”

UCF should be unobservable in a 2-mm crystal!

Quantum diffusion

our xtal

Taking typical 2D LT = 1 µm at 1 K,

For systems size L > LT, consider (L/LT)d systems of size LT, UCF suppressed as

For AB oscillation, assuming 60 nm rings, N-1/2

~ 10-8

Size Scales

2/2)/( dT LL

TkDL

DL

BT

in

/

heGmeasured /~ 2

Quasi-periodic fluctuations vs T

Fluctuation falls off quickly with temperature

For UCF, expect slow power law decay ~T-1/4 or T-1/2

AB, AAS effect exponential in LT/P

Doesn’t match!

Non-Metallic Samples in High Field

Fluctuation does not change character significantly in enhanced field

Next Approach: Micro Samples

Micro Samples Cont.

Sample is gate-able

SdH signal not seen in 10 nm thick metallic sample

Exploring Callan-Harvey effect in a cleaved crystal

x

(b)

(c)

(a)

Desperately seeking Majorana bound state

SC1 SC2

(x)Majorana bound state

= 0 =

Open at byProximity effect

Surface topological states Fu and Kane, PRL 08

bound state wfelectron creation oper.

Neutral fermion that is its own anti-particle

Majorana oper.

Gap “twist” trapsMajorana)]()()()([ 2,1

2/*2,1

2/2,1 xcxexcxedx ii

Cu Doping: Intercalation between Layers

Hor et al., arXiv 0909.2890

Intercalation of Cu between layers

Confirmed by c-axis lattice parameter increase and STM data

Crystal quality checked by X-ray diffraction and electron diffraction

Diamagnetic Response at low T

Typical M for type II: -1000 A / m

From M(H), κ ~ 50

χ ~ -0.2

Impurity phases not SC above 1.8 K (Cu2Se, CuBi3Se5, Cu1.6Bi4.8Se8....)

Small deviations from Se stoichiometry suppress SC

Cu Doping: Transport Properties

Not complete resistive transition

Up to 80% transition has been seen

Carrier density relatively high

Upper Critical Field HC2

HC2 anisotropy moderate

ξc = 52 Å , ξab = 140 Å

HC2 estimate by extrapolation

Similar shape for H||ab

Ca Doping: Conclusions

Ca doping can bring samples from n-type to p-type

Non-metallic samples at threshold between the two reveal new transport properties

G ~ ln(H) at low H

δG ~ e2/h, quasiperiodic

Hard to fit with mesoscopic interpretation

No LL quantization seen up to 32 T

Metallic nanoscale samples show no LL

Summary

Doping of Bi2Se3 creates surprising effects

Ca doping: Quantum Corrections to Transport become strong

Cu Doping: Superconductivity

Next stage:

1.nm-thick gated, cleaved crystals

2.Proximity effect and Josephson current expt

END